A comparative study of one-step and multi-step numerical methods for solving ordinary differential equations in water tank drainage systems

Abstract

Numerical methods are essential for solving differential equations in applications such as water drainage systems, where precise water level control is critical for industrial and environmental processes. This study compares one-step numerical methods naming explicit Euler, implicit Euler, implicit midpoint, modified Euler, and fourth-order Runge-Kutta (RK4) with multi-step numerical methods, including Adams-Bashforth, Adams-Moulton, and Predictor-corrector schemes, to solve ordinary differential equations for water tank drainage systems. The analysis focuses on accuracy, stability, computational efficiency, and optimal step size selection. MATLAB scripts and Python (Google Colab) were used to evaluate each method's performance by calculating local and global errors, with detailed analyses of error versus step size, error versus computational effort, and computational effort versus step size. The results reveal that multi-step numerical methods provide superior accuracy and stability for long-term simulations but require greater memory resources, whereas one-step numerical methods are computationally faster but sensitive to step size selection, significantly influencing solution accuracy. This study offers practical recommendations for selecting numerical methods based on application-specific requirements, providing insights into optimizing numerical approaches for systems requiring precise water level control and balancing accuracy with computational efficiency.